






"""simple docstring"""





from __future__ import annotations

from collections.abc import Callable

_SCREAMING_SNAKE_CASE   :							Tuple          =		list[list[float | int]]







def 	lowerCamelCase__			(				_lowerCamelCase			:						Matrix						,							_lowerCamelCase			:						Matrix					) ->							Matrix:
  lowerCamelCase_       	=		len(_lowerCamelCase					)
  lowerCamelCase_       	=		[[0 for _ in range(size + 1					)] for _ in range(_lowerCamelCase					)]
  lowerCamelCase_       	=		42
  lowerCamelCase_       	=		42
  lowerCamelCase_       	=		42
  lowerCamelCase_       	=		42
  lowerCamelCase_       	=		42
  lowerCamelCase_       	=		42

  for row in range(_lowerCamelCase					):
    for col in range(_lowerCamelCase					):
      lowerCamelCase_       	=		matrix[row][col]

    lowerCamelCase_       	=		vector[row][0]

  lowerCamelCase_       	=		0
  lowerCamelCase_       	=		0
  while row < size and col < size:
    # pivoting
    lowerCamelCase_       	=		max((abs(augmented[rowa][col]					), rowa) for rowa in range(_lowerCamelCase						,							_lowerCamelCase					)					)[
        1
    ]
    if augmented[pivot_row][col] == 0:
      col += 1
      continue
    else:
      lowerCamelCase_	,      lowerCamelCase_       	=		augmented[pivot_row], augmented[row]

    for rowa in range(row + 1						,							_lowerCamelCase					):
      lowerCamelCase_       	=		augmented[rowa][col] / augmented[row][col]
      lowerCamelCase_       	=		0
      for cola in range(col + 1						,							size + 1					):
        augmented[rowa][cola] -= augmented[row][cola] * ratio

    row += 1
    col += 1

  # back substitution
  for col in range(1						,							_lowerCamelCase					):
    for row in range(_lowerCamelCase					):
      lowerCamelCase_       	=		augmented[row][col] / augmented[col][col]
      for cola in range(_lowerCamelCase						,							size + 1					):
        augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
  return [
      [round(augmented[row][size] / augmented[row][row]						,							10					)] for row in range(_lowerCamelCase					)
  ]







def 	lowerCamelCase__			(				_lowerCamelCase			:						list[int]					) ->							Callable[[int], int]:
  lowerCamelCase_       	=		len(_lowerCamelCase					)
  lowerCamelCase_       	=		[[0 for _ in range(_lowerCamelCase					)] for _ in range(_lowerCamelCase					)]
  lowerCamelCase_       	=		[[0] for _ in range(_lowerCamelCase					)]
  lowerCamelCase_       	=		42
  lowerCamelCase_       	=		42
  lowerCamelCase_       	=		42
  lowerCamelCase_       	=		42

  for x_val, y_val in enumerate(_lowerCamelCase					):
    for col in range(_lowerCamelCase					):
      lowerCamelCase_       	=		(x_val + 1) ** (size - col - 1)
    lowerCamelCase_       	=		y_val

  lowerCamelCase_       	=		solve(_lowerCamelCase						,							_lowerCamelCase					)

  def interpolated_func(_lowerCamelCase			:						int					) -> int:
    return sum(
        round(coeffs[x_val][0]					) * (var ** (size - x_val - 1))
        for x_val in range(_lowerCamelCase					)					)

  return interpolated_func







def 	lowerCamelCase__			(				_lowerCamelCase			:						int					) ->							int:
  return (
      1
      - variable
      + variable**2
      - variable**3
      + variable**4
      - variable**5
      + variable**6
      - variable**7
      + variable**8
      - variable**9
      + variable**10
  )







def 	lowerCamelCase__			(				_lowerCamelCase			:						Callable[[int], int] = question_function						,							_lowerCamelCase			:						int = 10					) ->							int:
  lowerCamelCase_       	=		[func(_lowerCamelCase					) for x_val in range(1						,							order + 1					)]

  lowerCamelCase_       	=		[
      interpolate(data_points[:max_coeff]					) for max_coeff in range(1						,							order + 1					)
  ]

  lowerCamelCase_       	=		0
  lowerCamelCase_       	=		42
  lowerCamelCase_       	=		42

  for poly in polynomials:
    lowerCamelCase_       	=		1
    while func(_lowerCamelCase					) == poly(_lowerCamelCase					):
      x_val += 1

    ret += poly(_lowerCamelCase					)

  return ret


if __name__ == "__main__":
  print(F'''{solution() = }''')












"""simple docstring"""





from sympy import diff, lambdify, symbols
from sympy.functions import *  # noqa: F403







def 	lowerCamelCase__			(				_lowerCamelCase			:						str						,							_lowerCamelCase			:						complex						,							_lowerCamelCase			:						str = "x"						,							_lowerCamelCase			:						float = 10**-10						,							_lowerCamelCase			:						int = 1						,							) ->							complex:
  lowerCamelCase_       	=		symbols(_lowerCamelCase					)
  lowerCamelCase_       	=		lambdify(_lowerCamelCase						,							_lowerCamelCase					)
  lowerCamelCase_       	=		lambdify(_lowerCamelCase						,							diff(_lowerCamelCase						,							_lowerCamelCase					)					)

  lowerCamelCase_       	=		starting_point

  while True:
    if diff_function(_lowerCamelCase					) != 0:
      lowerCamelCase_       	=		prev_guess - multiplicity * func(_lowerCamelCase					) / diff_function(
          _lowerCamelCase					)
    else:
      raise ZeroDivisionError('Could not find root'					) from None

    # Precision is checked by comparing the difference of consecutive guesses
    if abs(next_guess - prev_guess					) < precision:
      return next_guess

    lowerCamelCase_       	=		next_guess


# Let's Execute
if __name__ == "__main__":
  # Find root of trigonometric function
  # Find value of pi
  print(F'''The root of sin(x) = 0 is {newton_raphson('sin(x)', 2)}''')

  # Find root of polynomial
  # Find fourth Root of 5
  print(F'''The root of x**4 - 5 = 0 is {newton_raphson('x**4 -5', 0.4 +5j)}''')

  # Find value of e
  print(
      '''The root of log(y) - 1 = 0 is ''',
      F'''{newton_raphson('log(y) - 1', 2, variable='y')}''',
  )

  # Exponential Roots
  print(
      '''The root of exp(x) - 1 = 0 is''',
      F'''{newton_raphson('exp(x) - 1', 10, precision=0.005)}''',
  )

  # Find root of cos(x)
  print(F'''The root of cos(x) = 0 is {newton_raphson('cos(x)', 0)}''')




